1. Field of the Invention
This invention relates to calculating failure probabilities and more particularly to a methodology that involves identifying the most-probable-point in the original X-space with an iteration procedure.
2. Description of the Related Art
Several methods of approximation for calculating failure probabilities have been developed. These methods of approximation are described in the following references:
Ang, A. H.-S., and W. H. Tang, Probability Concepts in Engineering Planning and Design, II--Decision, Risk, and Reliability, John Wiley & Sons, Inc., New York, N.Y. 1984. PA1 Madsen, H. O., S. Krenk, and N. C. Lind, Methods of Structural Safety, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1986. PA1 Der Kiureghian, A., H.-Z. Lin, and S. J. Hwang, "Second-Order Reliability Applicatons," Journal of Engineering Mechanics, Vol. 113, No.8, ASCE, 1987, pp. 1208-1255. PA1 Khalessi, M. R., Y.-T. Wu, and T. Y. Torng, "A new Most-Probable-Point Search Procedure for Efficient Structural Reliability Analysis," Proceedings of the 32nd Structures, Structural Dynamics, and Materials Conference, Part 2, AIAA/ASME/ASCE/AHS/ASC, April 1991, pp. 1295-1304 PA1 Breitung, K., "Asymptotic Approximations for Multinormal Integrals," Journal of Engineering Mechanics, Vol. 110, No. 3, ASCE, 1984, pp. 357-366.
A common algorithm in these methods involves the transformation from the original X-space to a standard, uncorrelated normal U-space. This is conducted by the probability transformation that preserves cumulative density functions (CDFs) and PDFs at the linearization points in the iteration procedure. In reliability analysis, it is necessary to carry out this probability transformation only at a finite number of linearization points. The well-known first-and-second-order reliability methods (FORM/SORM) replace the limit-state function with a tangent plane and a quadratic surface, respectively, at a point u* on the limit-state surface, known as the U-space MPP (most-probable-point), which is at the minimum distance from the origin in the standard normal space.
The advantage of FORM/SORM analyses is that the major contribution to the failure probability comes from the vicinity of u* because the probability density decays exponentially with the distance from u*. However, because the probability transformation used in the conventional FORM/SORM analyses preserves the CDF at the linearization points, the U-space MPP is inconsistent with the true MPP, x*, which has the highest probability density function (PDF) in the failure domain, when one of the distributions of the basic variables is asymmetric. That is, by inverse probability transformation, the mapping of the U-space MPP, u*, in the original space is not the true MPP, x*. This can be easily shown from the fact that the origin in the U-space, which has the highest PDF in the standard normal space, is consistent with the median point instead of the mode point in the original space for asymmetric distributions.
Moreover, a nonlinearity from the probability transformation is imposed in the limit-state function in the standard normal space. This results in extra iterations to identify the MPP if probability transformation is employed in the analysis. Therefore, for accuracy and efficiency, it is desirable to develop a methodology to identify the MPP, x*, without using the probability transformation, and using x* to calculate the reliability.